Abstract

The normalized irradiance variance of a Gaussian beam in the weak-fluctuation regime is examined numerically with the use of various spectral models for refractive index fluctuations. The Kolmogorov, Tatarskii, and modified von Kármán spectral models are chosen as conventional models, while the Hill numerical spectral model and the Andrews analytic approximation to the Hill model are selected to feature the characteristic bump at high wave numbers. The latter two models are known to predict higher scintillation levels than conventional spectral models when the Fresnel zone and the inner scale are of comparable size. Outer scale effects appear minimal near the centerline of the beam but can reduce off-axis scintillation significantly. Inner scale effects are prominent on axis as well as off axis, although they sometimes tend to diminish near the diffractive beam edge. Analytic approximations are developed for the irradiance variance based on the Kolmogorov, modified von Kármán, and Andrews spectral models. These analytic expressions are generally in excellent agreement with numerical results.

© 1994 Optical Society of America

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  1. R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
    [CrossRef]
  2. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
    [CrossRef]
  3. L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
    [CrossRef]
  4. L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1960).
  5. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).
  6. D. L. Fried, J. B. Seidman, “Laser beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 181–185 (1967).
    [CrossRef]
  7. T. L. Ho, “Log-amplitude fluctuations of a laser beam in a turbulent atmosphere,” J. Opt. Soc. Am. 59, 385–390 (1969).
    [CrossRef]
  8. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” in Effective Utilization and Application of Small Format Camera Systems, F. R. LaGesse, ed., Proc. Soc. Photo-Opt. Instrum. Eng.58, 1523–1545 (1970).
  9. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 790–809 (1975).
  10. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 1669–1692 (1975).
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.
  12. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” in Optics in Solar Energy Utilization I, Y. H. Katz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.68, 1424–1443 (1980).
  13. V. E. Zuev, Laser Beams in the Atmosphere, S. Wood, trans. (Consultants Bureau, New York, 1982).
    [CrossRef]
  14. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
    [CrossRef]
  15. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex paraxial optical systems,” Appl. Opt. 32, 5918–5929 (1993).
    [CrossRef] [PubMed]
  16. E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azart, “Optical measurement of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
    [CrossRef]
  17. L. C. Andrews, “Consequences of a new refractive index spectral model on optical measurements of the inner scale,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scatter and Propagation, Florence, Italy, August 1991.
  18. A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect of irradiance variance measured for weak-to-strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
    [CrossRef]
  19. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
    [CrossRef]
  20. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).
  21. L. C. Andrews, R. L. Phillips, B. K. Shivamoggi, “Relations of the parameters of the I–Kdistribution for irradiance fluctuations to physical parameters of the turbulence,” Appl. Opt. 27, 2150–2156 (1988).
    [CrossRef] [PubMed]

1994 (1)

1993 (3)

1992 (1)

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

1988 (2)

1978 (2)

1969 (1)

1967 (1)

Andrews, L. C.

L. C. Andrews, W. B. Miller, J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
[CrossRef]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
[CrossRef]

L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex paraxial optical systems,” Appl. Opt. 32, 5918–5929 (1993).
[CrossRef] [PubMed]

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

L. C. Andrews, R. L. Phillips, B. K. Shivamoggi, “Relations of the parameters of the I–Kdistribution for irradiance fluctuations to physical parameters of the turbulence,” Appl. Opt. 27, 2150–2156 (1988).
[CrossRef] [PubMed]

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).

L. C. Andrews, “Consequences of a new refractive index spectral model on optical measurements of the inner scale,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scatter and Propagation, Florence, Italy, August 1991.

Azart, Z.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azart, “Optical measurement of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Azoulay, E.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azart, “Optical measurement of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 790–809 (1975).

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1960).

Churnside, J. H.

Clifford, S. F.

Cochetti, F.

Consortini, A.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 1669–1692 (1975).

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” in Optics in Solar Energy Utilization I, Y. H. Katz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.68, 1424–1443 (1980).

Fried, D. L.

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 790–809 (1975).

Hill, R. J.

Ho, T. L.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.

Jetter, A.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azart, “Optical measurement of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Kohnle, A.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azart, “Optical measurement of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” in Effective Utilization and Application of Small Format Camera Systems, F. R. LaGesse, ed., Proc. Soc. Photo-Opt. Instrum. Eng.58, 1523–1545 (1970).

Miller, W. B.

Phillips, R. L.

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 790–809 (1975).

Ricklin, J. C.

Seidman, J. B.

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 790–809 (1975).

Shivamoggi, B. K.

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” in Effective Utilization and Application of Small Format Camera Systems, F. R. LaGesse, ed., Proc. Soc. Photo-Opt. Instrum. Eng.58, 1523–1545 (1970).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).

Thiermann, V.

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azart, “Optical measurement of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Zuev, V. E.

V. E. Zuev, Laser Beams in the Atmosphere, S. Wood, trans. (Consultants Bureau, New York, 1982).
[CrossRef]

Appl. Opt. (2)

J. Fluid Mech. (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

J. Mod. Opt. (1)

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Phys. D (1)

E. Azoulay, V. Thiermann, A. Jetter, A. Kohnle, Z. Azart, “Optical measurement of the inner scale of turbulence,” J. Phys. D 21, 541–544 (1988).
[CrossRef]

Other (10)

L. C. Andrews, “Consequences of a new refractive index spectral model on optical measurements of the inner scale,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scatter and Propagation, Florence, Italy, August 1991.

L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1960).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (McGraw-Hill, New York, 1961).

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” in Effective Utilization and Application of Small Format Camera Systems, F. R. LaGesse, ed., Proc. Soc. Photo-Opt. Instrum. Eng.58, 1523–1545 (1970).

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 790–809 (1975).

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” in Guided Optical Communications, F. L. Thiel, ed., Proc. Soc. Photo-Opt. Instrum. Eng.63, 1669–1692 (1975).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” in Optics in Solar Energy Utilization I, Y. H. Katz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.68, 1424–1443 (1980).

V. E. Zuev, Laser Beams in the Atmosphere, S. Wood, trans. (Consultants Bureau, New York, 1982).
[CrossRef]

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).

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Figures (9)

Fig. 1
Fig. 1

Scaled scintillation index for a collimated beam (Ω0 = 1) with the use of the Kolmogorov, Tatarskii, and von Kármán spectra as a function of Ω. The lower curves denote on-axis (ρ = 0) values, and the upper curves correspond to ρ/W = 1. The path length is L = 250 m, the inner scale is l0 = 3.9 mm, and the outer scale is L0 = 1.7 m.

Fig. 2
Fig. 2

Same as Fig. 1 but for the Kolmogorov, Hill, and Andrews spectral models.

Fig. 3
Fig. 3

Scaled scintillation index for a convergent beam (Ω0 = 0.1) with the use of the Kolmogorov, Tatarskii, and von Kármán spectra as a function of Ω. The lower curves denote on-axis (ρ = 0) values, and the upper curves correspond to ρ/W = 1. The path length is L = 250 m, the inner scale is l0 = 3.9 mm, and the outer scale is L0 = 1.7m.

Fig. 4
Fig. 4

Same as Fig. 3 but for the Kolmogorov, Hill, and Andrews spectral models.

Fig. 5
Fig. 5

Scaled scintillation index for a collimated beam (Ω0 = 1) along the beam axis (ρ = 0) as a function of Ω with the use of the Kolmogorov and Andrews spectral models. Path length is fixed, and the cases Ql = 10 and Ql = 100 correspond to two different inner scale sizes.

Fig. 6
Fig. 6

Same as Fig. 5 but for a convergent beam with Ω0 = 0.1.

Fig. 7
Fig. 7

Scaled scintillation index for a collimated beam (Ω0 = 1) showing the approximate variance (dashed curves) from Table 1 and exact numerical results (solid curves) at the beam center and the diffractive beam edge for the Andrews bump spectrum. Path length is fixed, and the conditions are as given in Figs. 1 and 2.

Fig. 8
Fig. 8

Same as Fig. 7 but for a convergent beam with Ω0 = 0.1.

Fig. 9
Fig. 9

Irradiance variance with the use of the von Kármán spectrum (dashed curves) and the Andrews spectrum (solid curves) normalized by the irradiance variance calculated with the Kolmogorov spectrum as a function of Ql. Two beam sizes are illustrated over a fixed path.

Tables (2)

Tables Icon

Table 1 Approximate Expressions for Irradiance Variancea

Tables Icon

Table 2 Approximate Expressions for Irradiance Variancea

Equations (27)

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Φ n ( κ ) = 0.033 C n 2 exp ( - κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) 11 / 6 ,
Φ n ( κ ) = 0.033 C n 2 [ 1 + a 1 ( κ κ l ) - a 2 ( κ κ l ) 7 / 6 ] × exp ( - κ 2 / κ l 2 ) ( κ 2 + κ 0 2 ) 11 / 6 ,
Φ n ( κ ) = 0.033 C n 2 κ - 11 / 3 .
U ( ρ , 0 ) = exp [ - ( 1 W 0 2 + j k 2 R 0 ) ρ 2 ] ,
Ω 0 = 1 - L R 0 ,             Ω = 2 L k W 0 2
Θ = 1 + L R ,             Λ = 2 L k W 2 ,
Θ = Ω 0 Ω 0 2 + Ω 2 ,             Λ = Ω Ω 0 2 + Ω 2 .
σ I 2 ( ρ , L ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ L κ 2 ξ 2 / k ) × { I 0 ( 2 Λ ρ κ ξ ) - cos [ L κ 2 k ξ ( 1 - Θ ˜ ξ ) ] } d κ d ξ ,
Θ ˜ = 1 - Θ = - L R .
Q m = L κ m 2 k ,             Q l = L κ l 2 k ,             Q 0 = L κ 0 2 k ,
Ω 2.61 Ω 0 0.63 ,             0 < Ω 0 1 ,
Ω 0.38 Ω 0 1.37 ,             0 < Ω 0 1.
Ω 11 12 Ω 0 ,             0 < Ω 0 1.
σ I 2 ( ρ , L ) = σ I , r 2 ( ρ , L ) + σ I , l 2 ( L ) ,
σ I , r 2 ( ρ , L ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ L κ 2 ξ 2 / k ) × [ I 0 ( 2 Λ ρ ξ κ ) - 1 ] d κ d ξ ,
σ I , l 2 ( L ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ L κ 2 ξ 2 / k ) × { 1 - cos [ L κ 2 k ξ ( 1 - Θ ˜ ξ ) ] } d κ d ξ .
σ I , r 2 ( ρ , L ) = 0.132 π 2 C n 2 k 7 / 6 L 11 / 6 [ Γ ( - 5 / 6 ) ( 1 + Λ Q m Q m ) 5 / 6 × n = 1 ( - 5 / 6 ) n ( 1 / 2 ) n ( 1 ) n ( 3 / 2 ) n n ! ( 2 ρ 2 / W 2 ) n ( Λ Q m 1 + Λ Q m ) n × F 2 1 ( n - 5 6 , 1 ; n + 3 2 ; Λ Q m 1 + Λ Q m ) - 24 5 Λ Q 0 1 / 6 ρ 2 / W 2 ] ,
σ I , r 2 ( ρ , L ) 3.93 σ 1 2 Λ 5 / 6 [ ( Λ Q m 1 + 0.52 Λ Q m ) 1 / 6 - 1.29 ( Λ Q 0 ) 1 / 6 ] ρ 2 W 2 ,             ρ / W < 1.
σ I , l 2 ( L ) = 7.08 σ 1 2 Q m - 5 / 6 { Re [ n = 0 ( - 5 / 6 ) n ( 1 ) n ( 2 ) n n ! ( - j Q m ) n × F 2 1 ( - n , n + 1 ; n + 2 ; Θ ˜ + j Λ ) ] - F 2 1 ( - 5 6 , 5 2 ; 3 2 ; - Λ Q m ) } ,             Q m < 1 ,
F 2 1 ( - n , n + 1 ; n + 2 ; x ) ( 1 - 2 x / 3 ) n ,             x < 1 ,
n = 0 ( - 5 / 6 ) n ( 1 ) n ( 2 ) n n ! ( - j Q m ) n F 2 1 ( - n , n + 1 ; n + 2 ; Θ ˜ + j Λ ) 6 11 ( { 1 + Q m [ 2 Λ / 3 + j ( 1 + 2 Θ ) / 3 ] } 11 / 6 - 1 Q m [ 2 Λ / 3 + j ( 1 + 2 Θ ) / 3 ] ) ,
F 2 1 ( 1 - a , 1 ; 2 ; - x ) = ( 1 + x ) a - 1 a x .
F 2 1 ( - 5 6 , 5 2 ; 3 2 ; - Λ Q m ) ( 1 + 0.31 Λ Q m ) 5 / 6 ,
σ I , l 2 ( L ) 3.86 σ 1 2 { 0.40 [ ( 1 + 2 Θ ) 2 + ( 2 Λ + 3 / Q m ) 2 ] 11 / 12 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] 1 / 2 × sin ( 11 6 φ 1 + φ 2 ) - 6 Λ Q m 11 / 6 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] - 11 6 ( 1 + 0.31 Λ Q m Q m ) 5 / 6 } ,
φ 1 = tan - 1 [ ( 1 + 2 Θ ) Q m 3 + 2 Λ Q m ] ,             φ 2 = tan - 1 ( 2 Λ 1 + 2 Θ ) .
φ 3 = tan - 1 [ ( 1 + 2 Θ ) Q l 3 + 2 Λ Q l ] .
σ I 2 ( L ) = 3.86 σ 1 2 { 1 μ ( μ 2 + 1 Q m 2 ) 11 / 12 sin [ 11 6 tan - 1 ( μ Q m ) ] - 11 6 Q m - 5 / 6 } ,

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